Einstein Lie groups, geodesic orbit manifolds and regular Lie subgroups

Abstract

We study the relation between two special classes of Riemannian Lie groups [Formula: see text] with a left-invariant metric [Formula: see text]: The Einstein Lie groups, defined by the condition [Formula: see text], and the geodesic orbit Lie groups, defined by the property that any geodesic is the integral curve of a Killing vector field. The main results imply that extensive classes of compact simple Einstein Lie groups [Formula: see text] are not geodesic orbit manifolds, thus providing large-scale answers to a relevant question of Nikonorov. Our approach involves studying and characterizing the [Formula: see text]-invariant geodesic orbit metrics on Lie groups [Formula: see text] for a wide class of subgroups [Formula: see text] that we call (weakly) regular. By-products of our work are structural and characterization results that are of independent interest for the classification problem of geodesic orbit manifolds.